Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $p = \dfrac{3}{3(3a + 7)} \div \dfrac{2a}{9(3a + 7)} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{3}{3(3a + 7)} \times \dfrac{9(3a + 7)}{2a} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 3 \times 9(3a + 7) } { 3(3a + 7) \times 2a } $ $ p = \dfrac{27(3a + 7)}{6a(3a + 7)} $ We can cancel the $3a + 7$ so long as $3a + 7 \neq 0$ Therefore $a \neq -\dfrac{7}{3}$ $p = \dfrac{27 \cancel{(3a + 7})}{6a \cancel{(3a + 7)}} = \dfrac{27}{6a} = \dfrac{9}{2a} $